Chapter 10 Properties of Stock Options Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull

Notation Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull c: European call option price p: European put option price S0:S0: Stock price today K: Strike price T: Life of option :: Volatility of stock price C: American call option price P: American put option price ST:ST: Stock price at option maturity D: PV of dividends paid during life of option r Risk-free rate for maturity T with cont. comp.

Effect of Variables on Option Pricing (Table 10.1, page 215) Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 Variable cpCP S0S0 +−+− K −+−+ T ??++  ++++ r +−+− D −+−+ 3

American vs European Options Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull An American option is worth at least as much as the corresponding European option C  c P  p

Calls: An Arbitrage Opportunity? Suppose that Is there an arbitrage opportunity? Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull c = 3 S 0 = 20 T = 1 r = 10% K = 18 D = 0

Lower Bound for European Call Option Prices; No Dividends ( Equation 10.4, page 220) c  S 0 –Ke -rT Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull

Puts: An Arbitrage Opportunity? Suppose that Is there an arbitrage opportunity? Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull p = 1 S 0 = 37 T = 0.5 r =5% K = 40 D = 0

Lower Bound for European Put Prices; No Dividends (Equation 10.5, page 221) p  Ke -rT –S 0 Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull

Put-Call Parity: No Dividends Consider the following 2 portfolios: Portfolio A: European call on a stock + zero- coupon bond that pays K at time T Portfolio C: European put on the stock + the stock Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull

Values of Portfolios Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull S T > KS T < K Portfolio ACall option S T − K0 Zero-coupon bond KK Total STST K Portfolio CPut Option 0K− S T Share STST STST Total STST K

The Put-Call Parity Result (Equation 10.6, page 222) Both are worth max( S T, K ) at the maturity of the options They must therefore be worth the same today. This means that c + Ke -rT = p + S 0 Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull

Suppose that What are the arbitrage possibilities when p = 2.25 ? p = 1 ? Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull Arbitrage Opportunities c = 3 S 0 = 31 T = 0.25 r = 10% K =30 D = 0

Early Exercise Usually there is some chance that an American option will be exercised early An exception is an American call on a non- dividend paying stock This should never be exercised early Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull

An Extreme Situation For an American call option: S 0 = 100; T = 0.25; K = 60; D = 0 Should you exercise immediately? What should you do if You want to hold the stock for the next 3 months? You do not feel that the stock is worth holding for the next 3 months? Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull

Reasons For Not Exercising a Call Early (No Dividends) No income is sacrificed You delay paying the strike price Holding the call provides insurance against stock price falling below strike price Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull

Bounds for European or American Call Options (No Dividends) Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull

Should Puts Be Exercised Early ? Are there any advantages to exercising an American put when S 0 = 60; T = 0.25; r =10% K = 100; D = 0 Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull

Bounds for European and American Put Options (No Dividends) Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull

The Impact of Dividends on Lower Bounds to Option Prices (Equations 10.8 and 10.9, page 229) Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull

Extensions of Put-Call Parity American options; D = 0 S 0 − K < C − P < S 0 − Ke −rT Equation 10.7 p. 224 European options; D > 0 c + D + Ke −rT = p + S 0 Equation p. 230 American options; D > 0 S 0 − D − K < C − P < S 0 − Ke −rT Equation p. 230 Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull